Pore-scale physics of ice melting within unconsolidated porous media revealed by non-destructive magnetic resonance characterization

Melting of ice in porous media widely exists in energy and environment applications as well as extraterrestrial water resource utilization. In order to characterize the ice-water phase transition within complicated opaque porous media, we employ the nuclear magnetic resonance (NMR) and imaging (MRI) approaches. Transient distributions of transverse relaxation time T2 from NMR enable us to reveal the substantial role of inherent throat and pore confinements in ice melting among porous media. More importantly, the increase in minimum T2 provides new findings on how the confinement between ice crystal and particle surface evolves inside the pore. For porous media with negligible gravity effect, both the changes in NMR-determined melting rate and our theoretical analysis of melting front confirm that conduction is the dominant heat transfer mode. The evolution of mushy melting front and 3D spatial distribution of water content are directly visualized by a stack of temporal cross-section images from MRI, in consistency with the corresponding NMR results. For heterogeneous porous media like lunar regolith simulant, the T2 distribution shows two distinct pore size distributions with different pore-scale melting dynamics, and its maximum T2 keeps increasing till the end of melting process instead of reaching steady in homogeneous porous media.


S2 Phase Change Heat Transfer Analysis
To further understand the heat transfer mechanism during ice melting in porous media, a one-dimensional analysis for the phase change process is conducted by considering temperature change in the radial direction only without heat generation.As shown by the radial cross-section of porous media in Fig. 5(c) & 5(d) in the main manuscript, the annular region contains water in porous matrix with an effective thermal conductivity of kwater,eff, while icing porous media remains around the center with an effective thermal conductivity of kice,eff as defined below:

 
, media 1 where ϕ is the porosity of the porous media.A sharp interface between solid and liquid phases is assumed with temperature of the ice is close to its melting temperature.During the melting process, both sensible and latent heat transfer takes place.Stefan number, which is the ratio of sensible to latent heat, is defined as 1 : where cp is the specific heat capacity of water, S is found to be 0.0627.As St < 0.1, the sensible heat transfer during the phase change process can be neglected 2 .With this assumption, the energy equations for ice and water regions (Fig. 5d in the main manuscript) are given by Eq. (S4) and Eq.(S5) respectively: By evaluating the variation of mf r as in Fig. 5d in the main manuscript, Eq. (S9) can be used to estimate the surface temperature o T , which increases slightly (within 3 degrees) during the whole melting process of porous media with 100-200 μm glass beads.This slight increase in temperature will reduce temperature difference between the porous sample and surrounding, eventually reducing melting rate in second stage of melting process (t > 28 min) as shown in Fig. 4d in the main manuscript.

S4 Characterization of Gravity Effect on Ice Melting by Temperature Distribution Measurements
Temperature distribution measurements for vertically oriented samples have been obtained, where the samples are slowly heated by a top surface at 5 o C while the side wall and bottom surface areas are thermally insulated.Temperature changes of porous media with different glass bead sizes and orientations are summarized in Table S2.Fig. S4a shows the representative temperature reading for ice melting in porous media with the glass beads of 1-50  under vertical orientation.It is shown that the melting front took 986 seconds to reach the top thermocouple TC-8, while it took 1024 and 1102 seconds to reach thermocouple TC-6, and TC-2, respectively.Fig. S4b shows the effect of pore size and sample orientation on time taken by the melting front to reach TC-2.The results clearly indicate that melting time increases with the increase in pore size for horizontally oriented samples.However, an opposite trend is observed for vertically orientated samples as increase in pore size reduces melting time.As shown in Table S2 and Fig. S4b, the results under vertical orientation indicate faster ice melting in large glass beads as compared to small glass beads.This means that the melting front was moving faster inside large pores than small pores, which shows that natural convection is the major heat transfer mode in the vertically oriented porous media.Along with the melting ice within the top part of the sample, the density increases because of water density inversion behavior, and gravity drives water to the bottom section to displace the colder ice.This happens at a relatively fast rate among the large-sized pores, because water has lower flow resistance in larger pores and natural convection enhances heat transfer besides conduction.As a measure of the fluid flow velocity across porous media, permeability has been recognized as the major factor that causes faster melting among large glass beads in vertical orientation 4 .For rocks, natural soil, and other heterogeneous porous media, permeability behaves anisotropically and exhibits different values along different directions.However, for homogenous and isotropic porous media, permeability is the same in all directions.The theoretical permeability is measured with the Carman Kozeny equation 5,6 while the falling head permeability method 7 is used for experimental permeability.
where   is the pore diameter, and γ is the liquid fraction.Larger pore size and liquid fraction usually result in higher permeability.Although Eq. (S10) is an empirical correlation, it gives good results for a porous medium that contains particles of spherical shape with a narrow size range.The equation is best suited to measure permeability during the phase change process as it can provide the permeability values throughout the melting process as a function of the liquid fraction.The value of the liquid fraction is zero at the beginning when it is in ice state, and then it increases to 1 when it is fully saturated with water.Therefore, the permeability must be initially zero, and then it keeps increasing.In porous media with large glass beads and pores, gravity has a strong impact and imposes high permeability.
The transient permeability is predicted throughout melting process by using the Carman-Kozeny equation, as seen in Fig. S4c.At the start, the liquid fraction is very small, and permeability is also small for all the porous media.This is due to the exponent of the liquid fraction, as it has a dominant effect in reducing the permeability.However, after t = 25 minutes it rises exponentially for higher size glass beads.It can be seen that the permeability is proportional to the pore size.For the tight porous media with the smallest bead size (1-50 ), the permeability reaches a maximum value of 0.41 Darcy, which shows the strong resistance of porous media for fluid flow.In addition, permeability is experimentally measured at a fully saturated state and compared with the predicted maximum permeability, where the maximum permeability refers to full saturation when the liquid fraction is 1.The comparison of results given in Fig. S4d shows that predicted values for permeability are in good agreement with the experiments.For instance, the measured permeability is 15.98 Darcy for glass beads with a size range of 100-200 μm while the predicted value is 14.83 Darcy.The permeability of loose porous media with large glass beads is higher than the permeability of tight porous media, which could be a reason for the higher melting rate in the large glass beads than the small glass beads under the vertical orientation.In general, it does show the significant impact of permeability in controlling and facilitating the movement of the ice-melting front in porous media.

sTT
is surface temperature of the NMR core, f T is melting or fusion temperature of ice and f h  is the latent heat of fusion.For s

Fig
Fig. S4.(a) Transient temperature distribution during ice melting in the tight porous medium with 1-50  glass (b) Effect of glass beads size on time taken by the melting front to reach thermocouple TC-2.Permeability of porous media made of different-sized glass beads; (c) Transient permeability calculated using Carmen-Kozeny equation; (d) Comparison of experimentally measured and predicted permeability at full saturation.

Table S1 :
Estimation of melting point depression (∆  ) of water confined in pore and throat of porous media based on NMR data

Table S2 :
Melting time of various icing porous media under vertical and horizontal orientations at locations TC-2 to TC-8